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Convergence of Numerical Solutions of Stochastic Differential Delay Equations / ULISES MUNOZ

Swansea University Author: ULISES MUNOZ

DOI (Published version): 10.23889/SUThesis.67952

Abstract

In this thesis we investigate explicit numerical approximations for stochastic differential delay equations (SDDEs) under a local Lipschitz condition by employing the adaptive Euler-Maruyama (EM) method. Working in both finite and infinite horizons, we achieve strong convergence results of the adaptive...

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Published: Swansea University, Wales, UK 2024
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
Supervisor: Yuan, C.
URI: https://cronfa.swan.ac.uk/Record/cronfa67952
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first_indexed 2024-10-10T11:31:55Z
last_indexed 2024-10-10T11:31:55Z
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spelling v2 67952 2024-10-10 Convergence of Numerical Solutions of Stochastic Differential Delay Equations aa5368c19fb0aa860186000b47648825 ULISES MUNOZ ULISES MUNOZ true false 2024-10-10 In this thesis we investigate explicit numerical approximations for stochastic differential delay equations (SDDEs) under a local Lipschitz condition by employing the adaptive Euler-Maruyama (EM) method. Working in both finite and infinite horizons, we achieve strong convergence results of the adaptive EM solution. We also obtain the order of convergence in finite horizon. In addition, we show almost sure exponential stability of the adaptive approximate solution for both SDEs and SDDEs. Further, we prove strong convergence of the adaptive solution for McKean-Vlasov SDDEs (MV-SDDEs). In the second part of the thesis, we estimate the variance of two coupled paths derived with the Multilevel Monte Carlo method combined with the EM discretization scheme for the simulation of MV-SDEs with small noise first and for MV-SDDEs later. The result often translates into a more efficient method than the standard Monte Carlo method combined with algorithms tailored to the small noise setting. E-Thesis Swansea University, Wales, UK Adaptive Euler Maruyama scheme; McKean-Vlasov Stochastic differential delay equations (MV-SDDEs); Strong convergence; Boundedness of the pth- moments; Almost sure exponential stability; Multilevel Monte Carlo simulation; Variance of two coupled paths 17 9 2024 2024-09-17 10.23889/SUThesis.67952 A selection of content is redacted or is partially redacted from this thesis to protect sensitive and personal information. COLLEGE NANME COLLEGE CODE Swansea University Yuan, C. Doctoral Ph.D 2024-10-10T12:43:31.8853473 2024-10-10T12:20:18.1049338 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics ULISES MUNOZ 1 67952__32581__b530a074b23e4309ab66f09d1670822d.pdf 2024_Munoz_U.final.67952.pdf 2024-10-10T12:30:38.2858599 Output 984192 application/pdf E-Thesis – open access true Copyright: The Author, Ulises Botija Munoz, 2024 true eng
title Convergence of Numerical Solutions of Stochastic Differential Delay Equations
spellingShingle Convergence of Numerical Solutions of Stochastic Differential Delay Equations
ULISES MUNOZ
title_short Convergence of Numerical Solutions of Stochastic Differential Delay Equations
title_full Convergence of Numerical Solutions of Stochastic Differential Delay Equations
title_fullStr Convergence of Numerical Solutions of Stochastic Differential Delay Equations
title_full_unstemmed Convergence of Numerical Solutions of Stochastic Differential Delay Equations
title_sort Convergence of Numerical Solutions of Stochastic Differential Delay Equations
author_id_str_mv aa5368c19fb0aa860186000b47648825
author_id_fullname_str_mv aa5368c19fb0aa860186000b47648825_***_ULISES MUNOZ
author ULISES MUNOZ
author2 ULISES MUNOZ
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publishDate 2024
institution Swansea University
doi_str_mv 10.23889/SUThesis.67952
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
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description In this thesis we investigate explicit numerical approximations for stochastic differential delay equations (SDDEs) under a local Lipschitz condition by employing the adaptive Euler-Maruyama (EM) method. Working in both finite and infinite horizons, we achieve strong convergence results of the adaptive EM solution. We also obtain the order of convergence in finite horizon. In addition, we show almost sure exponential stability of the adaptive approximate solution for both SDEs and SDDEs. Further, we prove strong convergence of the adaptive solution for McKean-Vlasov SDDEs (MV-SDDEs). In the second part of the thesis, we estimate the variance of two coupled paths derived with the Multilevel Monte Carlo method combined with the EM discretization scheme for the simulation of MV-SDEs with small noise first and for MV-SDDEs later. The result often translates into a more efficient method than the standard Monte Carlo method combined with algorithms tailored to the small noise setting.
published_date 2024-09-17T12:43:30Z
_version_ 1812527170517467136
score 11.030737

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